MONOCHROMATIC SUMS AND QUOTIENTS NEAR ZERO

Source abstract

Abstract Goswami [‘Monochromatic translated product and answering Sahasrabudhe’s conjecture’, Adv. Combin. (2026), https://doi.org/10.19086/aic.2026.2] proved that whenever the set N $\mathbb N$ double struck upper N of natural numbers is finitely coloured, the set { a , b , a b , b ( a + 1 ) } $\{a, b, ab, b(a+1)\}$ StartSet a comma b comma a b comma b left parenthesis a plus 1 right parenthesis EndSet is monochromatic. This also established a variant of the long-standing Hindman’s conjecture, which asks for a monochromatic set of the form { a , b , a b , a + b } $\{a, b, ab, a+b\}$ StartSet a comma b comma a b comma a plus b EndSet , and disproved a conjecture proposed by Sahasrabudhe that { a , b , a ( b + 1 ) } $\{a, b, a(b + 1)\}$ StartSet a comma b comma a left parenthesis b plus 1 right parenthesis EndSet is not partition regular. In this paper, we prove that { a , b , a b , b ( a + 1 ) } $\{a, b, ab, b(a+1)\}$ StartSet a comma b comma a b comma b left parenthesis a plus 1 right parenthesis EndSet is monochromatic near zero, which means that for every finite colouring of a dense subsemigroup of ( ( 0 , ∞ ) , + ) $((0, \infty ), +)$ left parenthesis left parenthesis 0 comma infinity right parenthesis comma plus right parenthesis , the set { a , b , a b , b ( a + 1 ) } $\{a, b, ab, b(a+1)\}$ StartSet a comma b comma a b comma b left parenthesis a plus 1 right parenthesis EndSet is monochromatic near zero. In other words, given a dense subsemigroup of ( ( 0 , ∞ ) , + ) $((0, \infty ), +)$ left parenthesis left parenthesis 0 comma infinity right parenthesis comma plus right parenthesis with a finite colouring, we can find a , b $a, b$ a comma b in the subsemigroup as small as we want such that the set { a , b , a b , b ( a + 1 ) } $\{a, b, ab, b(a+1)\}$ StartSet a comma b comma a b comma b left parenthesis a plus 1 right parenthesis EndSet is monochromatic. We also show that the pattern x , y , x + y , y / x $x, y, x+y, {y}/{x}$ x comma y comma x plus y comma y divided by x is partition regular near zero.